The purpose of this research study is to learn if adding hydroxychloroquine (HCQ) to erlotinib helps treat non-small cell lung cancer (NSCLC). Another goal of this research study is to learn more about NSCLC and how it may respond to study treatment. Erlotinib (Tarceva) is a type of drug called a tyrosine kinase inhibitor (TKI). TKIs block a protein called the epidermal growth factor receptor (EGFR). EGFR may control tumor growth and tumor cell survival. However, although TKI drugs can work for some lung cancer patients for a period of time, eventually the tumor finds a way to resist or counteract the TKI treatment and it begins to grow again. Hydroxychloroquine (HCQ) is a drug approved by the FDA for treating malaria, rheumatoid arthritis, and several other diseases. Laboratory research suggests that when HCQ is given with a TKI, it may help delay or prevent TKI resistance from developing.